If m = 9/25, w = 15/32, and m + w + c = 1, which of the following gives...

1. Translate the problem requirements: We have three values m, w, and c that sum to 1. We're given \(\mathrm{m} = \frac{9}{25}\) and \(\mathrm{w} = \frac{15}{32}\), and we need to find c, then arrange all three values from smallest to largest.
2. Convert fractions to comparable forms: Convert the given fractions to decimals or find a common way to compare them easily.
3. Calculate the missing value: Use the constraint \(\mathrm{m} + \mathrm{w} + \mathrm{c} = 1\) to solve for c.
4. Order the three values: Compare the decimal values to determine which is smallest, middle, and largest.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know and what we need to find:

We have three numbers: m, w, and c

• \(\mathrm{m} = \frac{9}{25}\) (given)
• \(\mathrm{w} = \frac{15}{32}\) (given)
• c = unknown (we need to find this)
• All three numbers add up to 1: \(\mathrm{m} + \mathrm{w} + \mathrm{c} = 1\)

Our goal is to arrange these three numbers from smallest to largest and match that order to one of the answer choices.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Convert fractions to comparable forms

To compare fractions easily, let's convert both given fractions to decimals:

For \(\mathrm{m} = \frac{9}{25}\):
Since \(25 \times 4 = 100\), we can write: \(\frac{9}{25} = \frac{9 \times 4}{25 \times 4} = \frac{36}{100} = 0.36\)

For \(\mathrm{w} = \frac{15}{32}\):
Let's divide: \(15 \div 32 = 0.46875\)
We can round this to 0.47 for easy comparison, but let's keep 0.46875 for accuracy.

So we have:

• m = 0.36
• w = 0.46875 (approximately 0.47)

3. Calculate the missing value

Now we can find c using the equation \(\mathrm{m} + \mathrm{w} + \mathrm{c} = 1\):

\(\mathrm{c} = 1 - \mathrm{m} - \mathrm{w}\)
\(\mathrm{c} = 1 - 0.36 - 0.46875\)
\(\mathrm{c} = 1 - 0.82875\)
\(\mathrm{c} = 0.17125\)

Let's double-check this makes sense: \(0.36 + 0.46875 + 0.17125 = 1.00000\) ✓

So our three values are:

• m = 0.36
• w = 0.46875
• c = 0.17125

4. Order the three values

Now let's arrange our decimal values from smallest to largest:

• c = 0.17125 (smallest)
• m = 0.36 (middle)
• w = 0.46875 (largest)

Therefore, in increasing order: c, m, w

Looking at our answer choices, this matches choice A: c, m, w

Final Answer

The values in increasing order are c, m, w.

Answer: A

Verification: \(\mathrm{c} = 0.17125 < \mathrm{m} = 0.36 < \mathrm{w} = 0.46875\), and \(\mathrm{c} + \mathrm{m} + \mathrm{w} = 0.17125 + 0.36 + 0.46875 = 1\) ✓

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting "increasing order"
Students may confuse "increasing order" with "decreasing order" and arrange the values from largest to smallest instead of smallest to largest. This fundamental misunderstanding would lead them to select answer choice E (w, m, c) instead of the correct answer A (c, m, w).

Faltering Point 2: Attempting to compare fractions without converting to common form
Students might try to directly compare \(\frac{9}{25}\) and \(\frac{15}{32}\) without converting them to decimals or finding a common denominator, leading to incorrect ordering of m and w from the start.

Errors while executing the approach

Faltering Point 1: Decimal conversion errors
When converting \(\frac{15}{32}\) to decimal form, students may make division errors and get an incorrect value (such as 0.48 instead of 0.46875), which could affect the final ordering.

Faltering Point 2: Arithmetic mistakes when calculating c
Students may make computational errors when subtracting \(\mathrm{m} + \mathrm{w}\) from 1, such as: \(\mathrm{c} = 1 - 0.36 - 0.46875 = 0.18125\) instead of 0.17125, potentially changing the relative position of c in the ordering.

Faltering Point 3: Rounding errors affecting comparison
Students might round their decimal values too early in the process (like using 0.47 for w instead of 0.46875), which could lead to incorrect comparisons between closely valued numbers.

Errors while selecting the answer

Faltering Point 1: Matching the correct ordering to wrong answer choice
Even after correctly determining that the increasing order is c, m, w, students might hastily scan the answer choices and accidentally select a different option that looks similar, such as choice C (m, w, c) or choice B (c, w, m).